* Time Series ARIMA Models in SAS; * Copyright 2013 by Ani Katchova; proc import out= work.data datafile= "C:\Econometrics\Data\timeseries_ppi.csv" dbms=csv replace; getnames=yes; datarow=2; run; * Creating a differenced variable; data data; set data; dppi=dif(ppi); lppi=lag(ppi); ldppi=lag(dppi); run; %let %let %let %let %let %let
ylist = ppi; dylist = dppi; time = t; lylist = lppi; trend=trend; xlist = cpi gdp;
proc means data=data; var &ylist &dylist &time; run; * Plotting the data; proc gplot data=data; plot &ylist*&time; plot &dylist*&time; run; * ARIMA identification; proc arima data=data; identify var=&ylist stationarity=(adf); run; * Dickey-Fuller test regressions; proc reg data=data; model &dylist = &lylist; model &dylist = &lylist &trend; run; * ARIMA for differenced variable; proc arima data=data; identify var=&ylist(1) stationarity=(adf); run;
* ARIMA(1,0,0) or AR(1); proc arima data=data; identify var=&ylist; estimate p=1 method=ml; run;
* ARIMA(2,0,0) or AR(2); proc arima data=data; identify var=&ylist; estimate p=2; run; * ARIMA(0,0,1) or MA(1); proc arima data=data; identify var=&ylist; estimate q=1; run; * ARIMA(1,0,1) or ARMA(1,1); proc arima data=data; identify var=&ylist; estimate p=1 q=1; run; * ARIMA(1,1,0); proc arima data=data; identify var=&dylist; estimate p=1; run; * ARIMA(0,1,1); proc arima data=data; identify var=&dylist; estimate q=1; run; * ARIMA(1,1,1); proc arima data=data; identify var=&dylist; estimate p=1 q=1; run; * ARIMA(1,1,3); proc arima data=data; identify var=&dylist; estimate p=1 q=3; run; * ARIMA(2,1,3); proc arima data=data; identify var=&dylist; estimate p=2 q=3; run; * ARIMA(2,0,1) with independent variables; proc arima data=data; identify var=&ylist crosscorr=(&xlist); estimate input=(&xlist) p=2 q=1 plot; run;
* ARIMA (1,0,1) forecasting; proc arima data=data; identify var=&ylist; estimate p=1 q=1; forecast lead=12; run; * ARIMA (1,1,1) forecasting; proc arima data=data; identify var=&dylist; estimate p=1 q=1; forecast lead=12; run;
The SAS System The MEANS Procedure Variable
N
Mean
Std Dev
Minimum
Maximum
ppi
169
64.6815385
30.2659545
25.2400000
110.4300000
dppi
168
0.4642857
0.9207450
-3.2100000
3.0800010
t
169
670564715
386030145
7862400.00
1333238400
The SAS System The ARIMA Procedure Name of Variable = ppi Mean of Working Series 64.68154 Standard Deviation
30.17628
Number of Observations
169
Autocorrelation Check for White Noise Autocorrelations
To Lag Chi-Square DF Pr > ChiSq 6
960.86
6
<.0001 0.990 0.978 0.966 0.952 0.937 0.923
12
1789.38 12
<.0001 0.908 0.894 0.880 0.866 0.852 0.838
18
2489.96 18
<.0001 0.824 0.810 0.795 0.780 0.765 0.749
24
3048.92 24
<.0001 0.732 0.716 0.698 0.681 0.663 0.645
Augmented Dickey-Fuller Unit Root Tests Type Zero Mean
Single Mean
Trend
Lags
Rho Pr < Rho
Tau Pr < Tau
0
0.9750
0.9071
5.66
0.9999
1
0.9132
0.8965
2.47
0.9969
2
0.8823
0.8908
2.10
0.9916
F Pr > F
0 -0.1024
0.9513 -0.26
0.9272 21.27 0.0010
1 -0.3804
0.9346 -0.51
0.8853
4.88 0.0425
2 -0.4918
0.9269 -0.61
0.8643
3.96 0.0911
0 -1.4094
0.9819 -0.79
0.9635
0.32 0.9900
1 -4.9336
0.8221 -1.45
0.8407
1.08 0.9570
2 -5.4808
0.7807 -1.47
0.8348
1.13 0.9503
The SAS System The REG Procedure Model: MODEL1 Dependent Variable: dppi Number of Observations Read
169
Number of Observations Used
168
Number of Observations with Missing Values
1
Analysis of Variance DF
Source Model
1
Error
Sum of Squares
Mean F Value Pr > F Square
0.05661 0.05661
0.07 0.7970
166 141.52119 0.85254
Corrected Total 167 141.57780 Root MSE
0.92333 R-Square
Dependent Mean
0.46429 Adj R-Sq -0.0056
Coeff Var
0.0004
198.87096 Parameter Estimates Parameter Standard t Value Pr > |t| Estimate Error
Variable
DF
Intercept
1
0.50357
0.16827
2.99
0.0032
lppi
1 -0.00060951
0.00237
-0.26
0.7970
The SAS System The REG Procedure Model: MODEL1 Dependent Variable: dppi
The SAS System The REG Procedure Model: MODEL2 Dependent Variable: dppi Number of Observations Read
169
Number of Observations Used
168
Number of Observations with Missing Values
1
Analysis of Variance DF
Source Model
2
Error
Sum of Squares
Mean F Value Pr > F Square
0.54332 0.27166
0.32 0.7282
165 141.03448 0.85475
Corrected Total 167 141.57780 Root MSE
0.92453 R-Square
Dependent Mean
0.46429 Adj R-Sq -0.0082
Coeff Var
0.0038
199.12939 Parameter Estimates
Variable
DF Parameter Standard t Value Pr > |t| Estimate Error
Intercept
1
0.58114
0.19737
2.94
0.0037
lppi
1
-0.00839
0.01058
-0.79
0.4289
trend
1
0.00496
0.00657
0.75
0.4516
The SAS System The REG Procedure Model: MODEL2 Dependent Variable: dppi
The SAS System The ARIMA Procedure Name of Variable = ppi Period(s) of Differencing
1
Mean of Working Series
0.464286
Standard Deviation
0.918001
Number of Observations
168
Observation(s) eliminated by differencing
1
Autocorrelation Check for White Noise Autocorrelations
To Lag Chi-Square DF Pr > ChiSq 6
102.82
6
<.0001 0.553
0.335
0.319 0.216
0.086
0.153
12
106.35 12
<.0001 0.082 -0.078 -0.080 0.023 -0.008 -0.006
18
112.72 18
<.0001 0.112
0.069
0.048 0.039
0.084
0.077
24
117.91 24
<.0001 0.076
0.085
0.049 0.033
0.047
0.089
Augmented Dickey-Fuller Unit Root Tests Type Zero Mean
Single Mean
Trend
Lags
Rho Pr < Rho
Tau Pr < Tau
F Pr > F
0 -59.1439
<.0001 -5.97
<.0001
1 -45.1897
<.0001 -4.66
<.0001
2 -25.1469
0.0002 -3.27
0.0012
0 -74.3553
0.0013 -6.86
<.0001 23.53 0.0010
1 -64.7748
0.0013 -5.49
<.0001 15.08 0.0010
2 -38.7515
0.0013 -3.85
0.0031
0 -74.3509
0.0005 -6.84
<.0001 23.41 0.0010
1 -64.5966
0.0005 -5.47
<.0001 15.03 0.0010
2 -38.2834
0.0008 -3.81
0.0184
7.45 0.0010
7.51 0.0197
The SAS System The ARIMA Procedure Name of Variable = ppi Mean of Working Series 64.68154 Standard Deviation Number of Observations
30.17628 169
Autocorrelation Check for White Noise To Lag Chi-Square DF Pr > ChiSq 6
960.86
Autocorrelations
6
<.0001 0.990 0.978 0.966 0.952 0.937 0.923
12
1789.38 12
<.0001 0.908 0.894 0.880 0.866 0.852 0.838
18
2489.96 18
<.0001 0.824 0.810 0.795 0.780 0.765 0.749
24
3048.92 24
<.0001 0.732 0.716 0.698 0.681 0.663 0.645
Maximum Likelihood Estimation
Parameter Estimate Standard Error t Value MU AR1,1
Approx Lag Pr > |t|
64.26332
125.46111
0.51
0.6085
0
0.99964
0.0022454
445.21
<.0001
1
Constant Estimate
0.022905
Variance Estimate
1.070851
Std Error Estimate
1.034819
AIC
500.4044
SBC
506.6642
Number of Residuals
169
Correlations of Parameter Estimates Parameter
MU
AR1,1
MU
1.000
0.955
AR1,1
0.955
1.000
Autocorrelation Check of Residuals To Lag Chi-Square DF Pr > ChiSq 6
213.28
Autocorrelations
5
<.0001 0.642 0.480 0.474 0.397 0.293 0.342
12
259.75 11
<.0001 0.283 0.151 0.149 0.228 0.200 0.199
18
342.65 17
<.0001 0.296 0.264 0.250 0.244 0.287 0.281
24
430.76 23
<.0001 0.282 0.292 0.262 0.251 0.263 0.291
30
498.16 29
<.0001 0.239 0.232 0.228 0.282 0.200 0.219
Model for variable ppi Estimated Mean 64.26332 Autoregressive Factors Factor 1: 1 - 0.99964 B**(1)
The SAS System The ARIMA Procedure Name of Variable = ppi Mean of Working Series 64.68154 Standard Deviation Number of Observations
30.17628 169
Autocorrelation Check for White Noise To Lag Chi-Square DF Pr > ChiSq 6
960.86
Autocorrelations
6
<.0001 0.990 0.978 0.966 0.952 0.937 0.923
12
1789.38 12
<.0001 0.908 0.894 0.880 0.866 0.852 0.838
18
2489.96 18
<.0001 0.824 0.810 0.795 0.780 0.765 0.749
24
3048.92 24
<.0001 0.732 0.716 0.698 0.681 0.663 0.645
Warning: The model defined by the new estimates is unstable. The iteration process has
been terminated.
Warning: Estimates may not have converged. ARIMA Estimation Optimization Summary Estimation Method
Conditional Least Squares
Parameters Estimated
3
Termination Criteria
Maximum Relative Change in Estimates
Iteration Stopping Value
0.001
Criteria Value
1.016656
Maximum Absolute Value of Gradient
1971.95
R-Square Change from Last Iteration
0.442814
Objective Function
Sum of Squared Residuals
Objective Function Value
126.1675
Marquardt's Lambda Coefficient
1E-6
Numerical Derivative Perturbation Delta
0.001
Iterations
18
Warning Message
Estimates may not have converged. Conditional Least Squares Estimation
Parameter Estimate Standard Error t Value MU
Approx Lag Pr > |t|
26.24523
0.83602
31.39
<.0001
0
AR1,1
1.29638
0.06963
18.62
<.0001
1
AR1,2
-0.29638
0.07017
-4.22
<.0001
2
Constant Estimate
4.013E-7
Variance Estimate
0.760045
Std Error Estimate
0.871806
AIC
436.2045
SBC
445.5942
Number of Residuals
169
* AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter MU AR1,1
MU
AR1,1
AR1,2
1.000
-0.019
0.019
-0.019
1.000
-1.000
Correlations of Parameter Estimates Parameter AR1,2
MU
AR1,1
AR1,2
0.019
-1.000
1.000
Autocorrelation Check of Residuals To Lag Chi-Square DF Pr > ChiSq 6
91.79
Autocorrelations
4
<.0001 0.368 0.273 0.364 0.291 0.139 0.284
12
114.78 10
<.0001 0.228 0.051 0.070 0.204 0.129 0.098
18
159.96 16
<.0001 0.262 0.178 0.173 0.150 0.223 0.195
24
205.75 22
<.0001 0.193 0.222 0.177 0.166 0.177 0.238
30
238.76 28
<.0001 0.149 0.162 0.135 0.254 0.097 0.145
Model for variable ppi Estimated Mean 26.24523 Autoregressive Factors Factor 1: 1 - 1.29638 B**(1) + 0.29638 B**(2)
The SAS System The ARIMA Procedure Name of Variable = ppi Mean of Working Series 64.68154 Standard Deviation Number of Observations
30.17628 169
Autocorrelation Check for White Noise To Lag Chi-Square DF Pr > ChiSq 6
960.86
Autocorrelations
6
<.0001 0.990 0.978 0.966 0.952 0.937 0.923
12
1789.38 12
<.0001 0.908 0.894 0.880 0.866 0.852 0.838
18
2489.96 18
<.0001 0.824 0.810 0.795 0.780 0.765 0.749
24
3048.92 24
<.0001 0.732 0.716 0.698 0.681 0.663 0.645
Conditional Least Squares Estimation
Parameter Estimate Standard Error t Value
Approx Lag Pr > |t|
MU
63.10136
2.39720
26.32
<.0001
0
MA1,1
-0.93793
0.02740
-34.23
<.0001
1
Constant Estimate
63.10136
Variance Estimate
264.9198
Std Error Estimate
16.27636
AIC
1424.512
SBC
1430.772
Number of Residuals
169
* AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter
MU
MA1,1
MU
1.000
0.153
MA1,1
0.153
1.000
Autocorrelation Check of Residuals To Lag Chi-Square DF Pr > ChiSq 6
838.19
Autocorrelations
5
<.0001 0.862 0.972 0.849 0.940 0.829 0.906
12
1560.24 11
<.0001 0.808 0.873 0.786 0.842 0.765 0.811
18
2171.65 17
<.0001 0.743 0.782 0.721 0.751 0.696 0.719
24
2661.17 23
<.0001 0.668 0.686 0.639 0.651 0.609 0.616
30
3026.09 29
<.0001 0.577 0.581 0.546 0.545 0.513 0.509
Model for variable ppi Estimated Mean 63.10136 Moving Average Factors Factor 1: 1 + 0.93793 B**(1)
The SAS System The ARIMA Procedure Name of Variable = ppi Mean of Working Series 64.68154 Standard Deviation Number of Observations
30.17628 169
Autocorrelation Check for White Noise To Lag Chi-Square DF Pr > ChiSq 6
960.86
Autocorrelations
6
<.0001 0.990 0.978 0.966 0.952 0.937 0.923
12
1789.38 12
<.0001 0.908 0.894 0.880 0.866 0.852 0.838
18
2489.96 18
<.0001 0.824 0.810 0.795 0.780 0.765 0.749
24
3048.92 24
<.0001 0.732 0.716 0.698 0.681 0.663 0.645
Warning: The model defined by the new estimates is unstable. The iteration process has
been terminated.
Warning: Estimates may not have converged. ARIMA Estimation Optimization Summary Estimation Method
Conditional Least Squares
Parameters Estimated
3
Termination Criteria
Maximum Relative Change in Estimates
Iteration Stopping Value
0.001
Criteria Value
0.97506
Maximum Absolute Value of Gradient
1672.52
R-Square Change from Last Iteration
0.415385
Objective Function
Sum of Squared Residuals
Objective Function Value
132.5082
Marquardt's Lambda Coefficient
1E-6
Numerical Derivative Perturbation Delta
0.001
Iterations
12
Warning Message
Estimates may not have converged. Conditional Least Squares Estimation
Parameter Estimate Standard Error t Value
Approx Lag Pr > |t|
MU
26.23885
0.84942
30.89
<.0001
0
MA1,1
-0.31130
0.08578
-3.63
0.0004
1
AR1,1
1.00000
0.0019167
521.73
<.0001
1
Constant Estimate
1.628E-6
Variance Estimate
0.798242
Std Error Estimate
0.893444
AIC
444.4913
SBC
453.881
Number of Residuals
169
* AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter
MU
MA1,1
AR1,1
MU
1.000
0.030
0.007
MA1,1
0.030
1.000
0.253
Correlations of Parameter Estimates Parameter AR1,1
MU
MA1,1
AR1,1
0.007
0.253
1.000
Autocorrelation Check of Residuals To Lag Chi-Square DF Pr > ChiSq 6
117.10
Autocorrelations
4
<.0001 0.402 0.363 0.385 0.325 0.183 0.303
12
145.64 10
<.0001 0.237 0.093 0.099 0.210 0.154 0.126
18
199.65 16
<.0001 0.274 0.188 0.204 0.174 0.240 0.217
24
255.98 22
<.0001 0.217 0.240 0.199 0.197 0.194 0.258
30
298.38 28
<.0001 0.166 0.197 0.147 0.272 0.121 0.176
Model for variable ppi Estimated Mean 26.23885 Autoregressive Factors Factor 1: 1 - 1 B**(1) Moving Average Factors Factor 1: 1 + 0.3113 B**(1)
The SAS System The ARIMA Procedure Name of Variable = dppi Mean of Working Series 0.464286 Standard Deviation
0.918001
Number of Observations
168
Autocorrelation Check for White Noise Autocorrelations
To Lag Chi-Square DF Pr > ChiSq 6
102.82
6
<.0001 0.553
0.335
0.319 0.216
0.086
0.153
12
106.35 12
<.0001 0.082 -0.078 -0.080 0.023 -0.008 -0.006
18
112.72 18
<.0001 0.112
0.069
0.048 0.039
0.084
0.077
24
117.91 24
<.0001 0.076
0.085
0.049 0.033
0.047
0.089
Conditional Least Squares Estimation
Parameter Estimate Standard Error t Value
Approx Lag Pr > |t|
MU
0.45192
0.13171
3.43
0.0008
0
AR1,1
0.55487
0.06476
8.57
<.0001
1
Constant Estimate
0.201162
Variance Estimate
0.591422
Std Error Estimate
0.76904
AIC
390.5135
SBC
396.7614
Number of Residuals
168
* AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter MU AR1,1
MU
AR1,1
1.000
-0.018
-0.018
1.000
Autocorrelation Check of Residuals Autocorrelations
To Lag Chi-Square DF Pr > ChiSq 6
14.59
5
0.0123 -0.032 -0.056
0.169
0.095 -0.137
0.153
12
26.96 11
0.0047
0.094 -0.153 -0.109
18
33.26 17
0.0104
0.163
0.005
0.075
0.018
24
36.58 23
0.0360
0.013
0.066 -0.002 -0.014 -0.005
0.110
30
45.44 29
0.0267 -0.041
0.116 -0.032 -0.102
0.007 -0.036
0.007 -0.061
0.159 -0.111 -0.019
Model for variable dppi Estimated Mean 0.451917 Autoregressive Factors Factor 1: 1 - 0.55487 B**(1)
The SAS System The ARIMA Procedure Name of Variable = dppi Mean of Working Series 0.464286 Standard Deviation
0.918001
Number of Observations
168
Autocorrelation Check for White Noise Autocorrelations
To Lag Chi-Square DF Pr > ChiSq 6
102.82
6
<.0001 0.553
0.335
0.319 0.216
0.086
0.153
12
106.35 12
<.0001 0.082 -0.078 -0.080 0.023 -0.008 -0.006
18
112.72 18
<.0001 0.112
0.069
0.048 0.039
0.084
0.077
24
117.91 24
<.0001 0.076
0.085
0.049 0.033
0.047
0.089
Conditional Least Squares Estimation
Parameter Estimate Standard Error t Value MU MA1,1
Approx Lag Pr > |t|
0.46466
0.09139
5.08
<.0001
0
-0.48912
0.06889
-7.10
<.0001
1
Constant Estimate
0.464664
Variance Estimate
0.636118
Std Error Estimate
0.79757
AIC
402.753
SBC
409.0009
Number of Residuals
168
* AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter MU MA1,1
MU
MA1,1
1.000
-0.007
-0.007
1.000
Autocorrelation Check of Residuals Autocorrelations
To Lag Chi-Square DF Pr > ChiSq 6
27.59
5
<.0001
0.105
0.245
0.173
0.177 -0.037
12
31.46 11
0.0009
0.055 -0.071 -0.079
18
37.34 17
0.0030
0.146 -0.008
0.057 -0.012
24
41.85 23
0.0095
0.040
30
51.80 29
0.0057 -0.051
0.161
0.059 -0.005 -0.060 0.074
0.038
0.011
0.036 -0.012
0.123
0.069 -0.085
0.156 -0.096
0.032
0.068
Model for variable dppi Estimated Mean 0.464664 Moving Average Factors Factor 1: 1 + 0.48912 B**(1)
The SAS System The ARIMA Procedure Name of Variable = dppi Mean of Working Series 0.464286 Standard Deviation
0.918001
Number of Observations
168
Autocorrelation Check for White Noise Autocorrelations
To Lag Chi-Square DF Pr > ChiSq 6
102.82
6
<.0001 0.553
0.335
0.319 0.216
0.086
0.153
12
106.35 12
<.0001 0.082 -0.078 -0.080 0.023 -0.008 -0.006
18
112.72 18
<.0001 0.112
0.069
0.048 0.039
0.084
0.077
24
117.91 24
<.0001 0.076
0.085
0.049 0.033
0.047
0.089
Conditional Least Squares Estimation
Parameter Estimate Standard Error t Value
Approx Lag Pr > |t|
MU
0.43146
0.15913
2.71
0.0074
0
MA1,1
0.25590
0.13981
1.83
0.0690
1
AR1,1
0.72813
0.10089
7.22
<.0001
1
Constant Estimate
0.117299
Variance Estimate
0.589302
Std Error Estimate
0.76766
AIC
390.8951
SBC
400.267
Number of Residuals
168
* AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter
MU
MA1,1
AR1,1
1.000
-0.079
-0.094
MA1,1
-0.079
1.000
0.842
AR1,1
-0.094
0.842
1.000
MU
Autocorrelation Check of Residuals Autocorrelations
To Lag Chi-Square DF Pr > ChiSq 6
12.67
4
0.0130
0.022 -0.118
0.120
0.067 -0.144
0.137
12
27.53 10
0.0021
0.087 -0.184 -0.137
18
33.60 16
0.0062
0.155
0.025 -0.006 -0.044
0.074
0.023
24
36.80 22
0.0249
0.015
0.068 -0.005 -0.026
0.004
0.104
30
44.22 28
0.0264 -0.037 -0.014 -0.043
0.099 -0.040 -0.105
0.143 -0.108 -0.026
Model for variable dppi Estimated Mean 0.431456 Autoregressive Factors Factor 1: 1 - 0.72813 B**(1) Moving Average Factors Factor 1: 1 - 0.2559 B**(1)
The SAS System The ARIMA Procedure Name of Variable = dppi Mean of Working Series 0.464286 Standard Deviation
0.918001
Number of Observations
168
Autocorrelation Check for White Noise Autocorrelations
To Lag Chi-Square DF Pr > ChiSq 6
102.82
6
<.0001 0.553
0.335
0.319 0.216
0.086
0.153
12
106.35 12
<.0001 0.082 -0.078 -0.080 0.023 -0.008 -0.006
18
112.72 18
<.0001 0.112
0.069
0.048 0.039
0.084
0.077
24
117.91 24
<.0001 0.076
0.085
0.049 0.033
0.047
0.089
Conditional Least Squares Estimation
Parameter Estimate Standard Error t Value
Approx Lag Pr > |t|
MU
0.42181
0.16964
2.49
0.0139
0
MA1,1
0.24281
0.16925
1.43
0.1533
1
MA1,2
0.10888
0.11683
0.93
0.3527
2
MA1,3
-0.12407
0.10101
-1.23
0.2211
3
AR1,1
0.73735
0.15570
4.74
<.0001
1
Constant Estimate
0.110789
Variance Estimate
0.581535
Std Error Estimate
0.762585
AIC
390.6174
SBC
406.2373
Number of Residuals
168
* AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter MU
MU
MA1,1
MA1,2
MA1,3
AR1,1
1.000 -0.109 -0.065 -0.020 -0.115
MA1,1
-0.109
1.000
0.517
0.418
0.884
MA1,2
-0.065
0.517
1.000
0.251
0.686
MA1,3
-0.020
0.418
0.251
1.000
0.526
AR1,1
-0.115
0.884
0.686
0.526
1.000
Autocorrelation Check of Residuals Autocorrelations
To Lag Chi-Square DF Pr > ChiSq 6
6.40
2
0.0408
0.008 -0.015
12
16.92
8
0.0309
0.048 -0.153 -0.146
18
22.88 14
0.0623
0.152
0.007
0.066
0.031
24
26.35 20
0.1546
0.022
0.062 -0.012 -0.006 -0.012
0.114
30
35.87 26
0.0940 -0.063
0.024 -0.075
0.010
0.008
0.061 -0.120
0.134
0.052 -0.026 -0.091
0.024 -0.056
0.150 -0.118
Model for variable dppi Estimated Mean
0.42181
Autoregressive Factors Factor 1: 1 - 0.73735 B**(1) Moving Average Factors Factor 1: 1 - 0.24281 B**(1) - 0.10888 B**(2) + 0.12407 B**(3)
The SAS System The ARIMA Procedure Name of Variable = dppi Mean of Working Series 0.464286 Standard Deviation
0.918001
Number of Observations
168
Autocorrelation Check for White Noise Autocorrelations
To Lag Chi-Square DF Pr > ChiSq 6
102.82
6
<.0001 0.553
0.335
0.319 0.216
0.086
0.153
12
106.35 12
<.0001 0.082 -0.078 -0.080 0.023 -0.008 -0.006
18
112.72 18
<.0001 0.112
0.069
0.048 0.039
0.084
0.077
24
117.91 24
<.0001 0.076
0.085
0.049 0.033
0.047
0.089
Conditional Least Squares Estimation
Parameter Estimate Standard Error t Value
Approx Lag Pr > |t|
MU
0.43244
0.14612
2.96
0.0035
0
MA1,1
1.04754
0.18112
5.78
<.0001
1
MA1,2
-0.21287
0.14771
-1.44
0.1515
2
MA1,3
-0.32823
0.09138
-3.59
0.0004
3
AR1,1
1.51747
0.17595
8.62
<.0001
1
AR1,2
-0.71168
0.16096
-4.42
<.0001
2
Constant Estimate
0.083983
Variance Estimate
0.568152
Std Error Estimate
0.753759
AIC
387.6722
SBC
406.416
Number of Residuals
168
* AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter
MU
MA1,1
MA1,2
MA1,3
AR1,1
AR1,2
MU
1.000
0.039 -0.040 -0.006
0.048 -0.072
MA1,1
0.039
1.000 -0.818
0.906 -0.871
0.359
MA1,2
-0.040 -0.818
MA1,3
-0.006
0.359 -0.595
1.000
0.186 -0.038
AR1,1
0.048
0.906 -0.590
0.186
1.000 -0.946
AR1,2
-0.072 -0.871
1.000 -0.595 -0.590
0.621 -0.038 -0.946
0.621
1.000
Autocorrelation Check of Residuals Autocorrelations
To Lag Chi-Square DF Pr > ChiSq 6
6.64
1
0.0100
0.005 -0.002
12
13.27
7
0.0657
0.084 -0.090 -0.085
18
19.11 13
0.1196
0.151 -0.008
0.022 -0.062
24
22.08 19
0.2804
0.024
30
31.79 25
0.1643 -0.078
0.002
0.022 -0.136 0.095
0.137
0.003 -0.075 0.059
0.028
0.006
0.007 -0.011
0.096
0.009 -0.090
0.149 -0.103
0.025
0.071
Model for variable dppi Estimated Mean 0.432437 Autoregressive Factors Factor 1: 1 - 1.51747 B**(1) + 0.71168 B**(2) Moving Average Factors Factor 1: 1 - 1.04754 B**(1) + 0.21287 B**(2) + 0.32823 B**(3)
The SAS System The ARIMA Procedure Name of Variable = ppi Mean of Working Series 64.68154 Standard Deviation Number of Observations
30.17628 169
Autocorrelation Check for White Noise To Lag Chi-Square DF Pr > ChiSq 6
960.86
Autocorrelations
6
<.0001 0.990 0.978 0.966 0.952 0.937 0.923
12
1789.38 12
<.0001 0.908 0.894 0.880 0.866 0.852 0.838
18
2489.96 18
<.0001 0.824 0.810 0.795 0.780 0.765 0.749
24
3048.92 24
<.0001 0.732 0.716 0.698 0.681 0.663 0.645 Correlation of ppi and cpi
Variance of input = Number of Observations
1081.227 169
Correlation of ppi and gdp Variance of input = Number of Observations
3962848 169
Warning: The model defined by the new estimates is unstable. The iteration process has been terminated.
Warning: Estimates may not have converged. ARIMA Estimation Optimization Summary Estimation Method
Conditional Least Squares
Parameters Estimated
6
Termination Criteria
Maximum Relative Change in Estimates
Iteration Stopping Value Criteria Value
0.001 0.36856
Maximum Absolute Value of Gradient
254.7273
R-Square Change from Last Iteration
0.248226
Objective Function Objective Function Value
Sum of Squared Residuals 72.66322
ARIMA Estimation Optimization Summary Marquardt's Lambda Coefficient
1E-6
Numerical Derivative Perturbation Delta
0.001
Iterations
12
Warning Message
Estimates may not have converged. Conditional Least Squares Estimation
Parameter
Estimate Standard Error t Value
Approx Lag Variable Shift Pr > |t|
MU
6.88945
2.78279
2.48
0.0143
0 ppi
0
MA1,1
0.32485
0.35592
0.91
0.3628
1 ppi
0
AR1,1
1.51868
0.31257
4.86
<.0001
1 ppi
0
AR1,2
-0.51868
0.31350
-1.65
0.1000
2 ppi
0
NUM1
1.12185
0.10818
10.37
<.0001
0 cpi
0
NUM2
-0.0012860
0.0012160
-1.06
0.2918
0 gdp
0
Constant Estimate
4.796E-7
Variance Estimate
0.445787
Std Error Estimate
0.667673
AIC
348.9545
SBC
367.7339
Number of Residuals
169
* AIC and SBC do not include log determinant. Correlations of Parameter Estimates Variable Parameter
ppi MU
ppi MU
ppi MA1,1
ppi AR1,1
1.000 -0.221 -0.241
ppi AR1,2
cpi NUM1
gdp NUM2
0.238 -0.275 -0.707
ppi MA1,1 -0.221
1.000
0.980 -0.979 -0.009
0.218
ppi AR1,1 -0.241
0.980
1.000 -1.000 -0.031
0.253
ppi AR1,2
0.238 -0.979 -1.000
1.000
0.030 -0.249
cpi NUM1
-0.275 -0.009 -0.031
0.030
1.000 -0.453
gdp NUM2
-0.707
0.218
0.253 -0.249 -0.453
1.000
Autocorrelation Check of Residuals To Lag Chi-Square DF Pr > ChiSq
Autocorrelations
6
17.28
3
0.0006 0.212
12
21.72
9
0.0098 0.028 -0.114 -0.076 0.066
0.076
0.148 0.115 -0.042 0.108 0.023 0.008
Autocorrelation Check of Residuals To Lag Chi-Square DF Pr > ChiSq
Autocorrelations
18
32.52 15
0.0055 0.142
0.083
0.048 0.045
0.125 0.101
24
42.84 21
0.0033 0.080
0.148
0.063 0.015
0.057 0.130
30
47.68 27
0.0083 0.014
0.059
0.002 0.134 -0.024 0.038
Model for variable ppi Estimated Intercept 6.889449 Autoregressive Factors Factor 1: 1 - 1.51868 B**(1) + 0.51868 B**(2) Moving Average Factors Factor 1: 1 - 0.32485 B**(1) Input Number 1 Input Variable
cpi
Overall Regression Factor 1.121855 Input Number 2 Input Variable
gdp
Overall Regression Factor -0.00129
The SAS System The ARIMA Procedure Name of Variable = ppi Mean of Working Series 64.68154 Standard Deviation Number of Observations
30.17628 169
Autocorrelation Check for White Noise To Lag Chi-Square DF Pr > ChiSq 6
960.86
Autocorrelations
6
<.0001 0.990 0.978 0.966 0.952 0.937 0.923
12
1789.38 12
<.0001 0.908 0.894 0.880 0.866 0.852 0.838
18
2489.96 18
<.0001 0.824 0.810 0.795 0.780 0.765 0.749
24
3048.92 24
<.0001 0.732 0.716 0.698 0.681 0.663 0.645
Warning: The model defined by the new estimates is unstable. The iteration process has
been terminated.
Warning: Estimates may not have converged. ARIMA Estimation Optimization Summary Estimation Method
Conditional Least Squares
Parameters Estimated
3
Termination Criteria
Maximum Relative Change in Estimates
Iteration Stopping Value
0.001
Criteria Value
0.97506
Maximum Absolute Value of Gradient
1672.52
R-Square Change from Last Iteration
0.415385
Objective Function
Sum of Squared Residuals
Objective Function Value
132.5082
Marquardt's Lambda Coefficient
1E-6
Numerical Derivative Perturbation Delta
0.001
Iterations
12
Warning Message
Estimates may not have converged. Conditional Least Squares Estimation
Parameter Estimate Standard Error t Value
Approx Lag Pr > |t|
MU
26.23885
0.84942
30.89
<.0001
0
MA1,1
-0.31130
0.08578
-3.63
0.0004
1
AR1,1
1.00000
0.0019167
521.73
<.0001
1
Constant Estimate
1.628E-6
Variance Estimate
0.798242
Std Error Estimate
0.893444
AIC
444.4913
SBC
453.881
Number of Residuals
169
* AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter
MU
MA1,1
AR1,1
MU
1.000
0.030
0.007
MA1,1
0.030
1.000
0.253
Correlations of Parameter Estimates Parameter AR1,1
MU
MA1,1
AR1,1
0.007
0.253
1.000
Autocorrelation Check of Residuals To Lag Chi-Square DF Pr > ChiSq 6
117.10
Autocorrelations
4
<.0001 0.402 0.363 0.385 0.325 0.183 0.303
12
145.64 10
<.0001 0.237 0.093 0.099 0.210 0.154 0.126
18
199.65 16
<.0001 0.274 0.188 0.204 0.174 0.240 0.217
24
255.98 22
<.0001 0.217 0.240 0.199 0.197 0.194 0.258
30
298.38 28
<.0001 0.166 0.197 0.147 0.272 0.121 0.176
Model for variable ppi Estimated Mean 26.23885 Autoregressive Factors Factor 1: 1 - 1 B**(1) Moving Average Factors Factor 1: 1 + 0.3113 B**(1) Forecasts for variable ppi Obs Forecast Std Error 95% Confidence Limits 170 103.5671
0.8934
101.8160
105.3182
171 103.5671
1.4734
100.6793
106.4549
172 103.5671
1.8824
99.8777
107.2565
173 103.5671
2.2172
99.2214
107.9127
174 103.5671
2.5077
98.6521
108.4821
175 103.5671
2.7679
98.1421
108.9920
176 103.5671
3.0056
97.6762
109.4580
177 103.5671
3.2259
97.2444
109.8897
178 103.5671
3.4321
96.8404
110.2938
179 103.5671
3.6265
96.4592
110.6749
180 103.5671
3.8111
96.0975
111.0366
Forecasts for variable ppi Obs Forecast Std Error 95% Confidence Limits 181 103.5671
3.9871
95.7525
111.3816
The SAS System The ARIMA Procedure Name of Variable = dppi Mean of Working Series 0.464286 Standard Deviation
0.918001
Number of Observations
168
Autocorrelation Check for White Noise Autocorrelations
To Lag Chi-Square DF Pr > ChiSq 6
102.82
6
<.0001 0.553
0.335
0.319 0.216
0.086
0.153
12
106.35 12
<.0001 0.082 -0.078 -0.080 0.023 -0.008 -0.006
18
112.72 18
<.0001 0.112
0.069
0.048 0.039
0.084
0.077
24
117.91 24
<.0001 0.076
0.085
0.049 0.033
0.047
0.089
Conditional Least Squares Estimation
Parameter Estimate Standard Error t Value
Approx Lag Pr > |t|
MU
0.43146
0.15913
2.71
0.0074
0
MA1,1
0.25590
0.13981
1.83
0.0690
1
AR1,1
0.72813
0.10089
7.22
<.0001
1
Constant Estimate
0.117299
Variance Estimate
0.589302
Std Error Estimate
0.76766
AIC
390.8951
SBC
400.267
Number of Residuals
168
* AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter
MU
MA1,1
AR1,1
1.000
-0.079
-0.094
MA1,1
-0.079
1.000
0.842
AR1,1
-0.094
0.842
1.000
MU
Autocorrelation Check of Residuals Autocorrelations
To Lag Chi-Square DF Pr > ChiSq 6
12.67
4
0.0130
0.022 -0.118
0.120
0.067 -0.144
0.137
12
27.53 10
0.0021
0.087 -0.184 -0.137
18
33.60 16
0.0062
0.155
0.025 -0.006 -0.044
0.074
0.023
24
36.80 22
0.0249
0.015
0.068 -0.005 -0.026
0.004
0.104
30
44.22 28
0.0264 -0.037 -0.014 -0.043
0.099 -0.040 -0.105
0.143 -0.108 -0.026
Model for variable dppi Estimated Mean 0.431456 Autoregressive Factors Factor 1: 1 - 0.72813 B**(1) Moving Average Factors Factor 1: 1 - 0.2559 B**(1) Forecasts for variable dppi Obs Forecast Std Error 95% Confidence Limits 170
-0.4434
0.7677
-1.9480
1.0612
171
-0.2056
0.8490
-1.8695
1.4583
172
-0.0324
0.8890
-1.7749
1.7101
173
0.0937
0.9096
-1.6890
1.8765
174
0.1855
0.9203
-1.6182
1.9893
175
0.2524
0.9259
-1.5623
2.0671
176
0.3011
0.9289
-1.5195
2.1216
177
0.3365
0.9304
-1.4871
2.1602
178
0.3623
0.9313
-1.4629
2.1876
179
0.3811
0.9317
-1.4450
2.2073
180
0.3948
0.9319
-1.4318
2.2214
181
0.4048
0.9321
-1.4221
2.2316
